Let \(R\) be a commutative ring and denote by \(FI(R)\) the set of fuzzy ideals of \(R\); i.e. the set of functions \(\mu:R\to[0,1]\) with the following conditions: (i) \(\mu(x-y)\geq\min(\mu(x),\mu(y))\), (ii) \(\mu(x,y)\geq\max(\mu(x),\mu(y))\), and (iii) \(\mu(0)>0\). Define \(\mu(R)=\{\mu(x):x\in R\}\), \(\mu_t=\{x\in R:\mu(x)\geq t\}\) and \(C_\mu:=\{\mu_t:t\in\mu(R)\}\). The authors investigate the general problem of characterization of all those fuzzy ideals that can be identified with a given arbitrary family \(C\) of subsets of \(R\) together with a given arbitrary subset \(S\) of [0,1] and obtain two main results. Theorem 1. Let \(S\) be a subset of the interval [0,1] containing a maximal element \(\overline{s}>0\) and let \(C=\{I_t: t\in S\}\) be a decreasing chain of \(R\)-ideals indexed by \(S\). Then there exists a fuzzy ideal \(\mu\in FI(R)\) satisfying \(\mu(R)=S\) and \(C_{\mu}=C\) if and only if the following two conditions hold: (i) For every \(t\in S\), \(\bigcup\limits_{\tau\in(t,1]\cap S} I_{\tau}\varsubsetneqq I_t\), (ii) The ring \(R\) is the disjoint union \(R=\bigcup\limits_{t\in S} (I_t\backslash \bigcup\limits_{\tau\in(t,1]\cap S} I_{\tau})\). Given a subset \(S\) of the interval \([0,1]\) containing a maximal element \(\overline s>0\) and given a decreasing chain of \(R\)-ideals, \(C:=\{I_t:t\in S\}\), we let \(OI(S)\) denote the set of order isomorphisms from the partially ordered set \((S,\leq)\) onto itself. We also define \(B(C,S):=\{\mu\in FI(R):\mu(R)=S\text{ and } C_\mu=C\}\). Theorem 2. Assume the conditions of theorem 1 and that \(B(C,S)\neq\emptyset\). Then for every \(\mu_0\in B(C,S)\), we have \(B(C,S)=\{g\circ \mu_0: g\in OI(S)\}\), hence \(| B(C,S) |=| OI(S) |\). In particular, the pair \((C,S)\) determines a unique fuzzy ideal \(\mu\in B(C,S)\) if and only if \(| OI(S) |=1\), in which case \(OI(S)=\{\text{id}_S\}\), where \(\text{id}_S\) denotes the identity map on \(S\). Moreover, the authors give an open problem: What conditions, necessary and sufficient, must be satisfied by a partially ordered set \(S\) in order to have \(\mid OI(S)\mid=1\)?